A school cafeteria models the number of smoothies sold during week w of a semester with the function S defined...
GMAT Algebra : (Alg) Questions
A school cafeteria models the number of smoothies sold during week w of a semester with the function S defined below, where sales typically decline as students become busier with coursework.
\(\mathrm{S(w) = 310 - 12(w - 6)}\)
The variable w represents the week number of the semester (\(\mathrm{w = 1}\) for the first week, \(\mathrm{w = 2}\) for the second week, and so on). The constant 310 in this function estimates which of the following?
- The average weekly decrease in smoothies sold
- The difference in the number of smoothies sold between week 1 and week 6
- The number of smoothies sold during week 1
- The number of smoothies sold during week 6
1. TRANSLATE the problem information
- Given function: \(\mathrm{S(w) = 310 - 12(w - 6)}\)
- \(\mathrm{S(w)}\) represents smoothies sold during week w
- We need to determine what the constant 310 represents
2. INFER the approach to interpret the constant
- In the form \(\mathrm{S(w) = 310 - 12(w - 6)}\), the constant 310 stands alone
- To see what 310 represents, we need to find when the variable term \(\mathrm{-12(w - 6)}\) disappears
- This happens when \(\mathrm{(w - 6) = 0}\), or when \(\mathrm{w = 6}\)
3. SIMPLIFY by evaluating S(6)
- When \(\mathrm{w = 6}\): \(\mathrm{S(6) = 310 - 12(6 - 6)}\)
- \(\mathrm{S(6) = 310 - 12(0)}\)
\(\mathrm{S(6) = 310 - 0}\)
\(\mathrm{S(6) = 310}\) - Therefore, 310 smoothies are sold during week 6
4. Verify by expanding the function
- \(\mathrm{S(w) = 310 - 12(w - 6)}\)
\(\mathrm{S(w) = 310 - 12w + 72}\)
\(\mathrm{S(w) = 382 - 12w}\) - Check: \(\mathrm{S(6) = 382 - 12(6)}\)
\(\mathrm{S(6) = 382 - 72}\)
\(\mathrm{S(6) = 310}\) ✓
Answer: D
Why Students Usually Falter on This Problem
Most Common Error Path:
Weak INFER skill: Students may think the constant 310 represents the y-intercept (value when \(\mathrm{w = 0}\)) rather than understanding how to interpret constants in shifted linear functions.
They calculate \(\mathrm{S(0) = 310 - 12(0 - 6)}\)
\(\mathrm{S(0) = 310 - 12(-6)}\)
\(\mathrm{S(0) = 310 + 72}\)
\(\mathrm{S(0) = 382}\), then become confused because 382 isn't among the answer choices. This leads to confusion and guessing.
Second Most Common Error:
Poor TRANSLATE reasoning: Students may focus on the coefficient -12 and think it represents what the problem is asking about, confusing the rate of change with the constant term.
This may lead them to select Choice A (The average weekly decrease in smoothies sold) since -12 represents the weekly decrease.
The Bottom Line:
This problem requires understanding that constants in shifted linear functions have specific interpretations. The key insight is recognizing when the variable term equals zero to reveal what the constant represents in context.